Hermite-Hadamard Type Inequalities for Superquadratic Functions via Fractional Integrals
We use basic properties of superquadratic functions to obtain some new Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of existing results.
Let be a convex function and with ; then is known as the Hermite-Hadamard inequality.
This remarkable result is well known in the literature as the Hermite-Hadamard inequality. Recently, the generalizations, refinements, and improvements of the classical Hermite-Hadamard inequality have been the subject of intensive research.
Definition 1 (see ). A function is superquadratic provided that for all there exists a constant such that for all .
Theorem 2 (see ). The inequality holds for all probability measures and all nonnegative, -integrable functions , if and only if is superquadratic.
The discrete version of the above theorem is also used in the sequel.
Lemma 3 (see ). Suppose that is superquadratic. Let , , and let , where and . Then
Nonnegative superquadratic functions are much better behaved as we see next.
The Hermite-Hadamard inequalities for superquadratic functions are established by Banic et al. in .
Theorem 5 (see ). Let be a superquadratic function and ; then
It is remarkable that Sarikaya et al.  proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
Theorem 6 (see ). Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold: with .
We remark that the symbols and denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order with which are defined by respectively. Here, is the gamma function defined by .
Fractional integral operators are widely used to solve differential equations and integral equations. So a lot of work has been obtained on the theory and applications of fractional integral operators.
In this paper, we establish some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals which refine the inequalities of (6) for superquadratic functions which are also convex.
2. Main Results
Theorem 7. Let be a superquadratic integrable function on with . Then with .
Proof. By inequality (12), we get that By the change of variable , we get Therefore We have completed the proof.
Corollary 8. Putting in Theorem 7 gives
Let ; we get that Therefore, by replacing in (9) we get that
Theorem 9. Let be a superquadratic integrable function on with . Then with .
Proof. By inequality (20), we get that The proof is completed.
Corollary 10. Choosing in Theorem 9, one has
Corollary 11. Let be defined as in Theorem 7; one gets
Corollary 12. Taking in Corollary 11, one obtains
In this note, we obtain some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of known results. The concept of superquadratic functions in several variables is introduced in . An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for superquadratic functions in several variables via Riemann-Liouville integrals.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11).
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